What is the correct condition on \(a\) for the equation \(x^2-2(a+1)x+a^2+3=0\) to have real roots?
Answer and explanation
Correct answer: \(a\ge 1\)
For a quadratic equation to have real roots, its discriminant must satisfy \(D\ge 0\). Here, \(D=[-2(a+1)]^2-4(a^2+3)=8(a-1)\). Therefore, \(8(a-1)\ge 0\), which gives \(a\ge 1\). Option A incorrectly excludes \(a=1\), although at \(a=1\) the equation has equal real roots. Exam tip: For questions about real roots, begin by applying \(D\ge 0\).
Frequently asked questions
What is the correct answer to this question?
\(a\ge 1\)
Why is this the correct answer?
For a quadratic equation to have real roots, its discriminant must satisfy \(D\ge 0\). Here, \(D=[-2(a+1)]^2-4(a^2+3)=8(a-1)\). Therefore, \(8(a-1)\ge 0\), which gives \(a\ge 1\). Option A incorrectly excludes \(a=1\), although at \(a=1\) the equation has equal real roots. Exam tip: For questions about real roots, begin by applying \(D\ge 0\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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